Available-Case Restrictions for Conditional Distributions

Why the Conditioning Set Matters

An available-case restriction assigns a distribution to an outcome that is missing for some dropout patterns by borrowing the distribution observed among participants who have that outcome. The comparison is not uniquely defined until we say which participants are comparable. A pooled comparison across everyone observed at a visit can differ from a comparison among participants with the same baseline characteristics or the same observed prior outcomes.

This distinction matters because patterns can have different compositions. If participants who drop out are more concentrated in a particular baseline group or prior-outcome history, an unconditional available-case distribution may not represent the observed distribution for the relevant group. Conditioning changes the available cases that inform the missing outcome. It also imposes a stronger support requirement: there must be observed outcomes at the relevant conditioning values.

Definition: Let $Y_j$ be the outcome at visit $j$, and let $G$ denote the dropout pattern. Write $R_j=1$ when $Y_j$ is observed and $R_j=0$ when it is missing. The available-case set at visit $j$ consists of patterns with $R_j=1$. For a conditioning variable or history $C$, the pooled available-case conditional distribution is $$ F_j^A(y\mid c)=P(Y_j\le y\mid R_j=1,C=c). $$ An available-case restriction for a pattern $g$ with $R_j=0$ sets its missing-outcome distribution equal to this observed distribution: $$ P(Y_j\le y\mid G=g,C=c)=F_j^A(y\mid c). $$ This is a distributional assumption, not merely an equality of means.

For monotone dropout, $R_j=1$ indicates that the participant remained observed through visit $j$. The available-case distribution pools the patterns that have $Y_j$ observed; it does not necessarily equal the outcome distribution in any one such pattern. In particular, a restriction to the pooled available cases is weaker than requiring equality to every available pattern separately.

Conditional Restrictions and Their Assumptions

The conditioning set $C$ can be baseline information $X$, such as treatment group or a baseline measurement, or a prior outcome history that is observed for the pattern being considered. For example, for $Y_3$, a pattern that has $Y_1$ and $Y_2$ but not $Y_3$ can be compared at $C=(Y_1,Y_2)$. A pattern that dropped out before $Y_2$ cannot be conditioned on $Y_2$, because that history is unobserved for its participants.

The restriction requires several conditions to be meaningful. The outcome must be observed in the available cases used to estimate $F_j^A$; the conditioning information must be defined and observed for both the missing pattern and its comparison cases; and the relevant values of $C$ must have available cases. The equality of distributions for the missing pattern is an identifying assumption. Data on the missing outcomes cannot directly verify it.

Even where the restriction is specified, the observed conditional distribution may be estimated imprecisely if few available cases share a conditioning value. If there are no such cases, the conditional distribution is not empirically available there. Applying a parametric model or combining conditioning values may provide a prediction, but that adds assumptions rather than supplying the absent comparison.

How the Pooled Conditional Distribution Is Formed

The pooled available-case distribution is a mixture over the patterns that have the visit outcome observed. Conditional on $C=c$, its weights are the proportions of those available patterns among participants with $R_j=1$ and $C=c$. The mixture follows directly from conditioning on the pattern.

F_j^A(y\mid c) =\sum_{g:R_j(g)=1} P(Y_j\le y\mid G=g,C=c)\, P(G=g\mid R_j=1,C=c).

Here $R_j(g)$ is the observation indicator for pattern $g$. Because $Y_j$ is observed in every pattern in the sum, each conditional distribution on the right can be estimated from observed outcomes, subject to adequate data at $C=c$. The weights sum to one over available patterns. The restriction assigns this mixture to a pattern with $R_j=0$; it does not say that the missing pattern has the same distribution as each contributing pattern individually.

A short derivation makes the distinction clear. The events $G=g$ for patterns with $R_j(g)=1$ partition the event $R_j=1$. Therefore,

\begin{aligned} P(Y_j\le y\mid R_j=1,C=c) &=\sum_{g:R_j(g)=1} P(Y_j\le y,G=g\mid R_j=1,C=c)\\ &=\sum_{g:R_j(g)=1} P(Y_j\le y\mid G=g,R_j=1,C=c) P(G=g\mid R_j=1,C=c). \end{aligned}

Within each pattern, $R_j$ is fixed by the pattern, so the conditional outcome distribution in the second line is the pattern-specific observed distribution. The mixture identity is a probability rule; extending that observed mixture to a pattern where $Y_j$ is missing is the additional available-case restriction.

Worked Example: Conditioning on a Baseline Group

Suppose $Y_2$ is measured on a numerical scale and $X$ is a baseline group with values 0 and 1. At visit 2, patterns $G=1$ and $G=2$ have observed $Y_2$, while $G=3$ has missing $Y_2$. The hypothetical observed counts are:

For $X=0$, $G=1$ has 12 outcomes equal to 2 and 18 equal to 4; $G=2$ has 8 equal to 2 and 12 equal to 4.
For $X=1$, $G=1$ has 2 outcomes equal to 2 and 8 equal to 4; $G=2$ has 6 equal to 2 and 24 equal to 4.

Assume the available-case restriction conditional on $X$: within each baseline group, the distribution of $Y_2$ in $G=3$ equals the pooled observed distribution in $G=1$ and $G=2$. The needed outcomes and $X$ are observed in the comparison data, so the distributions can be calculated. For $X=0$, pooling gives 20 outcomes equal to 2 and 30 equal to 4, for 50 available cases. Thus,

P(Y_2=2\mid R_2=1,X=0)=\frac{20}{50}=0.400, \qquad P(Y_2=4\mid R_2=1,X=0)=\frac{30}{50}=0.600.

The conditional available-case mean is $(20(2)+30(4))/50=3.200$. Under the restriction, this is also the mean assigned to $G=3$ when $X=0$.

For $X=1$, the pooled counts are 8 outcomes equal to 2 and 32 equal to 4, out of 40. The probabilities are $8/40=0.200$ and $32/40=0.800$, and the mean is $(8(2)+32(4))/40=3.600$. The restriction therefore assigns a different conditional distribution to $G=3$ in the two baseline groups.

The observed counts establish each conditional distribution, but they do not establish that the unobserved $G=3$ outcomes follow it. That final equality is the identifying assumption.

Conditioning on Prior Outcomes

When $C$ is a prior outcome history, the restriction borrows from available cases with the same observed history, rather than from all participants who have the visit outcome. This can preserve differences in the distribution of $Y_j$ across histories. It also means that the available comparison may be narrow: only participants observed at visit $j$ with matching history contribute to that conditional distribution.

Worked Example: Borrowing Within Prior-Outcome History

Consider a hypothetical three-visit study. At visit 3, only the complete-pattern group $G=1$ has observed $Y_3$. Group $G=2$ has observed $Y_1,Y_2$ but missing $Y_3$. Let $C=Y_2$, with histories categorized as low or high. Among $G=1$, the low-history group has 10 participants, of whom 3 have $Y_3=1$; the high-history group has 20 participants, of whom 16 have $Y_3=1$. In $G=2$, 5 participants have low history and 15 have high history.

Assume the conditional available-case restriction for $G=2$: within each $Y_2$ category, its missing $Y_3$ distribution equals that observed in $G=1$. The prior outcome $Y_2$ is observed for $G=2$, and both history categories occur among available cases, so the comparison is defined. The available-case probabilities of $Y_3=1$ are $3/10=0.300$ for low history and $16/20=0.800$ for high history. The restriction assigns those probabilities to the corresponding $G=2$ histories.

The expected number of $Y_3=1$ outcomes assigned to the 20 members of $G=2$ is

$$ 5(0.300)+15(0.800)=1.5+12=13.5. $$

Thus the restricted mean proportion of $Y_3=1$ in $G=2$ is $13.5/20=0.675$. By contrast, the unconditional available-case proportion is $(3+16)/(10+20)=19/30=0.6333$. Using that one pooled proportion for every $G=2$ participant would disregard the different prior-history composition. The two answers differ because they encode different restrictions, not because one calculation is an arithmetic correction to the other.

Conditioning on more prior outcomes may be substantively attractive when those outcomes are informative about the missing measurement. But it narrows the comparison set and requires the history to be observed in the missing pattern. For each pattern, state the conditioning history that is actually available there; do not silently condition on an unobserved outcome.

Support, Diagnostics, and Alternatives

Worked Example: A History With No Available Comparison

Suppose the missing-$Y_3$ pattern $G=2$ has 6 participants with high $Y_2$, but none of the available cases with observed $Y_3$ has high $Y_2$. A restriction conditional on the exact value or category “high $Y_2$” cannot be estimated from those available cases: its conditional denominator is zero. There is no observed high-history distribution to pool, so the counts do not yield a numerical conditional probability for $G=2$ at that history.

The restriction is therefore unsupported at this conditioning value by the observed data. A fitted model could extrapolate from other histories, or the analyst could choose a coarser conditioning set, but either choice adds a modeling or identifying assumption. It would be incorrect to report an observed high-history available-case distribution that does not exist.

Useful diagnostics follow directly from the definition. Tabulate available-case counts across the conditioning values used for each missing pattern; check that the relevant histories are represented; and inspect how the pooled distribution changes across those histories. Very small counts, empty cells, or sharply changing estimates warn of unstable estimation or unsupported comparisons. These checks assess data support and estimation, not whether the assumed missing-outcome equality is true.

Conditional restrictions differ from two common alternatives. An unconditional available-case restriction uses $F_j^A(y)$ without conditioning, pooling across histories; it is simpler but can mask differences across $C$. A complete-case restriction instead uses a more limited complete-case comparison. Conditional available-case restrictions can use more of the observed data, but their results depend on the chosen conditioning set and require overlap at that set. None of these restrictions is automatically implied by observed data or by a missing-at-random label; each specifies how unobserved pattern-specific distributions are related to observed ones.

Common pitfall: “The outcome is observed for some participants, so their overall distribution can be assigned to every pattern with a missing outcome.” This reasoning ignores the conditioning set. If the restriction is conditional on baseline group or prior history, the comparison must use available cases at that same value. If none exist, the conditional restriction has no empirical comparison there; if they do exist, the equality for the missing pattern remains an assumption.

The practical choice is not simply “condition on as much as possible.” A conditioning set should be observed for the pattern being filled in, relevant to the outcome distribution, and supported by available cases. A coarse set may pool participants whose outcome distributions differ; a very fine set may leave too few or no comparisons. The resulting conditional available-case restriction is only as credible and estimable as that balance permits.

Check Your Understanding

Answer each question using the definitions and calculations in this tutorial.

At a visit, two patterns have observed outcomes and one pattern does not. Does the pooled available-case restriction require the missing pattern to match each observed pattern separately?
Why can a restriction conditional on $Y_2$ not be applied to a dropout pattern in which $Y_2$ is unobserved?
In the baseline-group example, what distribution is assigned to the missing pattern for $X=1$, and what is its mean?
What does a zero available-case count at a conditioning value imply for estimating that conditional available-case distribution?
Why can the unconditional and history-conditional available-case restrictions produce different assigned distributions?

Solutions. (1) No. It assigns the missing pattern the pooled mixture over available patterns, with weights given by their proportions among available cases at the conditioning value. (2) The conditioning value is not observed for that pattern, so the proposed conditional comparison is undefined for its participants. (3) It assigns probabilities 0.200 to $Y_2=2$ and 0.800 to $Y_2=4$, with mean $0.200(2)+0.800(4)=3.600$. (4) There is no observed conditional distribution at that value; a prediction requires additional modeling or a different conditioning choice. (5) The history-conditional restriction uses distributions within histories, whereas the unconditional restriction pools across histories; those distributions and the histories’ proportions can differ.